1
00:00:00,000 --> 00:00:08,870
CHRISTINE BREINER: Welcome
back to recitation.

2
00:00:08,870 --> 00:00:12,850
In this video, I'd like us to
practice integration by parts.

3
00:00:12,850 --> 00:00:16,410
Specifically, I'd like to solve
the following four problems.

4
00:00:16,410 --> 00:00:20,150
Or I'd like you to solve
the following four problems.

5
00:00:20,150 --> 00:00:21,760
I'd like us to find
antiderivatives

6
00:00:21,760 --> 00:00:25,200
for each of these functions.
x e to the minus x,

7
00:00:25,200 --> 00:00:29,340
x cubed over the quantity
1 plus x squared squared,

8
00:00:29,340 --> 00:00:33,540
arctan x, and natural
log x over x squared.

9
00:00:33,540 --> 00:00:36,300
And so the main goal, because
we're using integration

10
00:00:36,300 --> 00:00:38,780
by parts, is to figure out
what you should make u,

11
00:00:38,780 --> 00:00:40,970
and what you should
make v prime.

12
00:00:40,970 --> 00:00:42,611
And why don't you
give it a shot.

13
00:00:42,611 --> 00:00:43,860
Work on that for a little bit.

14
00:00:43,860 --> 00:00:45,620
I'm actually going
to give you one hint,

15
00:00:45,620 --> 00:00:47,830
and that's that this one,
you may want to break up

16
00:00:47,830 --> 00:00:49,140
in a nontraditional way.

17
00:00:49,140 --> 00:00:52,260
You may not want to break
it up as x cubed and 1

18
00:00:52,260 --> 00:00:53,350
over this function.

19
00:00:53,350 --> 00:00:56,930
You're going to want to split up
this function in the numerator.

20
00:00:56,930 --> 00:01:00,200
Part of it will be in u, part
of it will be in v prime.

21
00:01:00,200 --> 00:01:02,039
So that's my hint on number 2.

22
00:01:02,039 --> 00:01:04,580
So now with that information,
I'd like you to give it a shot,

23
00:01:04,580 --> 00:01:06,890
and then I'll come back, and
I'll show you how I do it.

24
00:01:14,420 --> 00:01:16,050
OK, welcome back.

25
00:01:16,050 --> 00:01:18,570
So again, what we're looking
for is antiderivatives for each

26
00:01:18,570 --> 00:01:21,180
of these four functions.

27
00:01:21,180 --> 00:01:22,820
And now, what I'm
going to do, is

28
00:01:22,820 --> 00:01:25,120
I'm going to help you
pick u and v prime,

29
00:01:25,120 --> 00:01:27,680
and then I'm going to show
you what answer I got.

30
00:01:27,680 --> 00:01:30,140
And I'm going to let you
do the work in the middle.

31
00:01:30,140 --> 00:01:32,890
So let's start
off with number 1.

32
00:01:35,960 --> 00:01:39,136
So if I have x e to the
minus x-- integral of x e

33
00:01:39,136 --> 00:01:43,370
to the minus x
dx, it's very easy

34
00:01:43,370 --> 00:01:46,600
to do either-- to make either
e to the minus x either u or v

35
00:01:46,600 --> 00:01:47,100
prime.

36
00:01:47,100 --> 00:01:48,330
Doesn't really matter.

37
00:01:48,330 --> 00:01:50,490
Because an integral
of e to the minus

38
00:01:50,490 --> 00:01:52,850
x is going to have an
e to the minus x again,

39
00:01:52,850 --> 00:01:55,550
and a derivative is going to
have an e to the minus x again,

40
00:01:55,550 --> 00:01:58,010
with a minus sign in
front, in both cases.

41
00:01:58,010 --> 00:01:59,740
But this doesn't really change.

42
00:01:59,740 --> 00:02:01,364
So we have-- when
we go up or down,

43
00:02:01,364 --> 00:02:03,155
it doesn't really matter
if we integrate up

44
00:02:03,155 --> 00:02:04,430
or take a derivative.

45
00:02:04,430 --> 00:02:06,270
So really it's, we
get to pick what

46
00:02:06,270 --> 00:02:08,160
we do with the e to the
minus x based on what

47
00:02:08,160 --> 00:02:09,610
we want to do with the x.

48
00:02:09,610 --> 00:02:12,310
Well, we like taking
derivatives of things

49
00:02:12,310 --> 00:02:16,710
that don't have two functions
of x, so it would be nice if we

50
00:02:16,710 --> 00:02:19,460
chose our integration
by parts pieces

51
00:02:19,460 --> 00:02:22,264
so that this thing
wasn't there anymore.

52
00:02:22,264 --> 00:02:24,180
So let me write down--
actually, before I even

53
00:02:24,180 --> 00:02:25,890
do number 1, maybe
I should remind you

54
00:02:25,890 --> 00:02:28,590
what the integration
by parts formula is.

55
00:02:28,590 --> 00:02:31,010
So let me just, I'll scratch
that out for a second.

56
00:02:31,010 --> 00:02:32,843
And what we're doing,
is we're going to have

57
00:02:32,843 --> 00:02:35,705
integral of u v prime dx.

58
00:02:35,705 --> 00:02:37,455
And if you recall what
you saw in lecture,

59
00:02:37,455 --> 00:02:42,880
is this should be equal to u*v
minus the integral v u prime

60
00:02:42,880 --> 00:02:44,240
dx.

61
00:02:44,240 --> 00:02:46,782
And we'll put that plus c,
because sometimes I forget

62
00:02:46,782 --> 00:02:47,740
to write it at the end.

63
00:02:47,740 --> 00:02:49,712
So I'll put it there,
so I don't forget.

64
00:02:49,712 --> 00:02:52,045
So really, what we're trying
to do, right, is pick the u

65
00:02:52,045 --> 00:02:53,540
and v prime.

66
00:02:53,540 --> 00:02:55,470
And so we want to
make this thing,

67
00:02:55,470 --> 00:02:58,030
this v u prime, as
simple as possible.

68
00:02:58,030 --> 00:03:00,700
So what I was saying is if we
make this v prime or this u,

69
00:03:00,700 --> 00:03:01,870
it doesn't matter.

70
00:03:01,870 --> 00:03:05,820
So let's pick whether we
want this to be u or v prime.

71
00:03:05,820 --> 00:03:09,930
Well, if I make this
u, then u prime is 1.

72
00:03:09,930 --> 00:03:10,900
That's good.

73
00:03:10,900 --> 00:03:14,450
If I make it v prime, then
v is x squared over 2.

74
00:03:14,450 --> 00:03:15,690
That's more complicated.

75
00:03:15,690 --> 00:03:18,480
So we obviously
want to make this u.

76
00:03:18,480 --> 00:03:23,150
So for number 1, we're going
to choose u is equal to x,

77
00:03:23,150 --> 00:03:28,520
and v prime is equal
to e to the minus x.

78
00:03:28,520 --> 00:03:30,650
And then you can
proceed from there.

79
00:03:30,650 --> 00:03:32,270
And I'll leave it at that.

80
00:03:32,270 --> 00:03:34,310
Well, actually, just
to make sure we're OK,

81
00:03:34,310 --> 00:03:36,970
I'll even write u
prime is equal to 1,

82
00:03:36,970 --> 00:03:40,750
and v is going to be equal
to negative e to the minus x.

83
00:03:40,750 --> 00:03:43,120
So we'd be able to
proceed from there, right?

84
00:03:43,120 --> 00:03:46,020
We have all the pieces we need.

85
00:03:46,020 --> 00:03:50,420
Now, number 2-- I'll give you
the final answers at the end.

86
00:03:50,420 --> 00:03:53,760
Number 2, picking u and v prime
is a little more complicated.

87
00:03:53,760 --> 00:03:56,920
And let's look at this function.

88
00:03:56,920 --> 00:04:01,436
x cubed over 1 plus
x squared squared.

89
00:04:01,436 --> 00:04:04,090
The problem with picking--
that does not look like a 2.

90
00:04:04,090 --> 00:04:04,660
Sorry.

91
00:04:04,660 --> 00:04:07,100
The problem with picking
u and v prime here,

92
00:04:07,100 --> 00:04:10,780
is that it's hard to see what's
going to be easy to integrate.

93
00:04:10,780 --> 00:04:13,700
So what we want to
do is rewrite this

94
00:04:13,700 --> 00:04:20,050
as-- let's see-- x
squared times x over 1

95
00:04:20,050 --> 00:04:24,840
plus x squared squared.

96
00:04:24,840 --> 00:04:27,324
And now, why is this any better?

97
00:04:27,324 --> 00:04:28,740
Well, I mean, it's
the same thing.

98
00:04:28,740 --> 00:04:31,370
But why does this help us
see what we want to do?

99
00:04:31,370 --> 00:04:35,770
Well, if you notice this thing
right here-- 1 plus x squared.

100
00:04:35,770 --> 00:04:37,020
What is its derivative?

101
00:04:37,020 --> 00:04:39,260
Its derivative is 2x.

102
00:04:39,260 --> 00:04:40,960
Up here we have an x.

103
00:04:40,960 --> 00:04:44,180
So this piece right
here looks like it

104
00:04:44,180 --> 00:04:47,730
could be much more easily
integrated than this right

105
00:04:47,730 --> 00:04:49,120
here.

106
00:04:49,120 --> 00:04:51,692
So this might be a
little counterintuitive,

107
00:04:51,692 --> 00:04:53,900
because we're going to take
the harder-looking thing,

108
00:04:53,900 --> 00:04:56,990
and make that our v prime.

109
00:04:56,990 --> 00:04:59,620
But the nice thing is
that we can actually

110
00:04:59,620 --> 00:05:01,340
integrate this quantity.

111
00:05:01,340 --> 00:05:07,170
So we choose, in this
case, this is our u,

112
00:05:07,170 --> 00:05:10,146
and this is our v prime.

113
00:05:10,146 --> 00:05:11,270
So how do I integrate this?

114
00:05:11,270 --> 00:05:13,660
Well, I integrate this
by using a substitution.

115
00:05:13,660 --> 00:05:16,260
And that will give me v.
And the derivative of this

116
00:05:16,260 --> 00:05:16,970
is quite simple.

117
00:05:16,970 --> 00:05:17,511
It's just 2x.

118
00:05:17,511 --> 00:05:18,610
Right?

119
00:05:18,610 --> 00:05:20,550
But this is the strategy
that we want here.

120
00:05:20,550 --> 00:05:23,400
Why did we even think to
split that up like that?

121
00:05:23,400 --> 00:05:25,710
Well, we knew we had to
deal with the denominator

122
00:05:25,710 --> 00:05:28,770
in some fashion, and
taking a derivative

123
00:05:28,770 --> 00:05:30,950
with this in the
denominator-- so

124
00:05:30,950 --> 00:05:34,040
putting this part of
the function in u--

125
00:05:34,040 --> 00:05:36,370
when I look at u prime,
it's going to be even worse.

126
00:05:36,370 --> 00:05:38,470
It's going to be a
higher power here.

127
00:05:38,470 --> 00:05:40,720
It's going to be a cubic
in the denominator.

128
00:05:40,720 --> 00:05:42,160
That's just making things worse.

129
00:05:42,160 --> 00:05:44,830
So we know we'd like to
integrate this denominator.

130
00:05:44,830 --> 00:05:46,870
We'd like it to be
a part of v prime.

131
00:05:46,870 --> 00:05:50,850
But the problem is that if I
put all the x cubed in the u,

132
00:05:50,850 --> 00:05:54,220
and if I just had a 1 here
for my v prime, that's, I

133
00:05:54,220 --> 00:05:55,820
can't really integrate
that very well.

134
00:05:55,820 --> 00:05:58,130
But if I keep one
of the x's, then I

135
00:05:58,130 --> 00:06:00,920
can integrate this quite
simply with a substitution.

136
00:06:00,920 --> 00:06:03,210
So that's the sort of
reasoning behind why

137
00:06:03,210 --> 00:06:04,850
we choose it that way.

138
00:06:04,850 --> 00:06:05,350
All right.

139
00:06:05,350 --> 00:06:07,183
We've got two more to
look at, and then I'll

140
00:06:07,183 --> 00:06:09,720
give you the answers.

141
00:06:09,720 --> 00:06:10,350
3.

142
00:06:10,350 --> 00:06:10,850
OK.

143
00:06:10,850 --> 00:06:13,410
3, you've seen
this trick before.

144
00:06:13,410 --> 00:06:16,500
The function was arctan x.

145
00:06:16,500 --> 00:06:18,620
Now, you've seen this
trick I'm about to do

146
00:06:18,620 --> 00:06:21,280
with natural log of x.

147
00:06:21,280 --> 00:06:23,170
The same kind of thing
with natural log of x.

148
00:06:23,170 --> 00:06:25,946
You actually saw this in
one of the lecture videos.

149
00:06:25,946 --> 00:06:27,570
Because there's only
one function here,

150
00:06:27,570 --> 00:06:30,710
you might think, well, I have
no idea what I'm supposed

151
00:06:30,710 --> 00:06:32,330
to pick for u and v prime.

152
00:06:32,330 --> 00:06:36,490
But remember, it's
really arctan x times 1.

153
00:06:36,490 --> 00:06:39,120
Now I have two functions.

154
00:06:39,120 --> 00:06:43,260
And what gives us a hint for
why we would want to do this,

155
00:06:43,260 --> 00:06:45,840
is that what's the
derivative of arctan x?

156
00:06:45,840 --> 00:06:47,197
Let me just remind you.

157
00:06:52,070 --> 00:06:56,410
d/dx of arctan is 1
over 1 plus x squared.

158
00:06:56,410 --> 00:06:57,970
Right?

159
00:06:57,970 --> 00:07:00,640
We're back to actually an
almost similar situation to what

160
00:07:00,640 --> 00:07:02,560
we had in the previous thing.

161
00:07:02,560 --> 00:07:04,990
d/dx of arctan x is 1
over 1 plus x squared.

162
00:07:04,990 --> 00:07:07,220
So taking a derivative
of this puts it

163
00:07:07,220 --> 00:07:09,600
in a form that almost
looks easy to integrate.

164
00:07:09,600 --> 00:07:11,920
What would make this
function easy to integrate?

165
00:07:11,920 --> 00:07:14,310
If there was an x up
here, instead of a 1.

166
00:07:14,310 --> 00:07:15,830
Then I could use substitution.

167
00:07:15,830 --> 00:07:18,090
Where do we get that
x from when we're

168
00:07:18,090 --> 00:07:20,880
solving this problem, where
we're actually finding

169
00:07:20,880 --> 00:07:22,930
an antiderivative of arctan x?

170
00:07:22,930 --> 00:07:27,070
Well, it's going to come from
the fact that I make this u,

171
00:07:27,070 --> 00:07:29,210
and I make 1 v prime.

172
00:07:29,210 --> 00:07:31,620
So let me write
that out explicitly.

173
00:07:31,620 --> 00:07:36,790
u I make arctan x,
and v prime I make 1.

174
00:07:36,790 --> 00:07:38,910
What does that do
in our formula?

175
00:07:38,910 --> 00:07:41,550
Well, we're going
to be integrating

176
00:07:41,550 --> 00:07:44,990
something that is v u prime.

177
00:07:44,990 --> 00:07:49,070
Well, v is going to be x, and
u prime we see right here.

178
00:07:49,070 --> 00:07:51,530
So it's going to be, I'm going
to be integrating x over 1

179
00:07:51,530 --> 00:07:54,280
plus x squared when I started
doing the integration by parts

180
00:07:54,280 --> 00:07:55,200
method.

181
00:07:55,200 --> 00:07:57,500
That's much simpler, as we
talked about previously,

182
00:07:57,500 --> 00:07:59,500
because the derivative
of x squared is 2x,

183
00:07:59,500 --> 00:08:01,440
and you have an x in
the numerator when

184
00:08:01,440 --> 00:08:03,250
you put in that v.

185
00:08:03,250 --> 00:08:07,630
So this is sort of the flavor
of how these things are actually

186
00:08:07,630 --> 00:08:08,475
working.

187
00:08:08,475 --> 00:08:12,410
So let me do the final one here.

188
00:08:12,410 --> 00:08:16,950
We have ln x over x squared.

189
00:08:16,950 --> 00:08:17,960
OK.

190
00:08:17,960 --> 00:08:20,240
Let me just tell you right now.

191
00:08:20,240 --> 00:08:22,500
In integration by
parts, natural log x

192
00:08:22,500 --> 00:08:25,139
is not something you
want to make the v prime.

193
00:08:25,139 --> 00:08:27,180
You don't want to try and
take an antiderivative.

194
00:08:27,180 --> 00:08:30,480
You know an antiderivative
of natural log of x.

195
00:08:30,480 --> 00:08:31,560
x ln x minus x.

196
00:08:31,560 --> 00:08:34,340
But that's certainly not going
to make things any easier.

197
00:08:34,340 --> 00:08:34,840
Right?

198
00:08:34,840 --> 00:08:36,214
You're actually--
then you've got

199
00:08:36,214 --> 00:08:38,460
a product of two
functions all of a sudden.

200
00:08:38,460 --> 00:08:40,700
Everything's getting
more complicated.

201
00:08:40,700 --> 00:08:43,640
But natural log of x has a very
nice derivative, because you

202
00:08:43,640 --> 00:08:46,300
end up with something that
has just a power of x.

203
00:08:46,300 --> 00:08:49,529
Derivative of natural log
of x is just 1 over x.

204
00:08:49,529 --> 00:08:51,070
So that's probably
the way you always

205
00:08:51,070 --> 00:08:54,150
want to go when you see natural
log of x in these integration

206
00:08:54,150 --> 00:08:56,650
by parts techniques.

207
00:08:56,650 --> 00:09:01,730
Because if I choose u is equal
to ln x, and then v prime.

208
00:09:01,730 --> 00:09:04,890
In this case, I'm going
to write it as a power.

209
00:09:04,890 --> 00:09:06,690
Let's think about what happened.

210
00:09:06,690 --> 00:09:09,050
u prime is 1 over x, right?

211
00:09:09,050 --> 00:09:11,500
So u prime is x to the minus 1.

212
00:09:11,500 --> 00:09:12,430
What's v?

213
00:09:12,430 --> 00:09:14,900
Well, it's something
like, let's see.

214
00:09:14,900 --> 00:09:16,215
Negative x to the minus 1.

215
00:09:16,215 --> 00:09:17,340
Something like that, right?

216
00:09:17,340 --> 00:09:18,714
Let's make sure
I did that right.

217
00:09:18,714 --> 00:09:20,410
Yeah, I think I did that right.

218
00:09:20,410 --> 00:09:23,550
So all of a sudden, if
I integrate v u prime,

219
00:09:23,550 --> 00:09:24,750
that's just a power rule.

220
00:09:24,750 --> 00:09:27,820
It's x to the minus 2,
negative x to the minus 2.

221
00:09:27,820 --> 00:09:31,120
So that's quite
easy to integrate.

222
00:09:31,120 --> 00:09:35,050
So again, when I see natural
log of x in an integration

223
00:09:35,050 --> 00:09:37,990
by parts method,
almost always, I

224
00:09:37,990 --> 00:09:39,770
hate to say always,
almost always,

225
00:09:39,770 --> 00:09:42,050
almost a guarantee that you
want to take a derivative.

226
00:09:42,050 --> 00:09:44,967
You want to make that the u.

227
00:09:44,967 --> 00:09:47,300
So hopefully that makes sense,
some of these strategies.

228
00:09:47,300 --> 00:09:49,760
I tried to pick ones that
were somewhat different,

229
00:09:49,760 --> 00:09:52,950
so you could see some different
types of strategies we needed.

230
00:09:52,950 --> 00:09:54,385
And now I've done these earlier.

231
00:09:54,385 --> 00:09:56,760
So I'm just going to write
down what the answers actually

232
00:09:56,760 --> 00:10:01,050
are, and you can
compare to what you got.

233
00:10:01,050 --> 00:10:03,064
So the answer to number
1, just to check.

234
00:10:09,770 --> 00:10:11,430
Number 2.

235
00:10:11,430 --> 00:10:13,200
Some of these are kind of long.

236
00:10:26,751 --> 00:10:27,250
Number 3.

237
00:10:36,910 --> 00:10:38,135
Number 4.

238
00:10:45,720 --> 00:10:46,930
So let's just go through.

239
00:10:46,930 --> 00:10:50,370
We get-- in number 1, we get
negative x e to the minus x

240
00:10:50,370 --> 00:10:52,930
minus e to the minus x plus c.

241
00:10:52,930 --> 00:10:54,740
Number 2, we get
negative x squared

242
00:10:54,740 --> 00:10:59,300
over 2 times 1 plus x squared
plus 1/2 natural log 1 plus x

243
00:10:59,300 --> 00:11:01,480
squared plus c.

244
00:11:01,480 --> 00:11:06,210
Three is x arc tan x minus 1/2
natural log of the quantity 1

245
00:11:06,210 --> 00:11:10,035
plus x squared plus c, and
four is negative natural log

246
00:11:10,035 --> 00:11:15,290
x over x minus 1 over x plus c.

247
00:11:15,290 --> 00:11:18,550
So again, the whole point of
this exercise, in my mind,

248
00:11:18,550 --> 00:11:21,600
is really to make sure we
get a good understanding of,

249
00:11:21,600 --> 00:11:24,440
when we're doing integration
by parts, which function makes

250
00:11:24,440 --> 00:11:27,200
the most sense to have as u, and
which function makes the most

251
00:11:27,200 --> 00:11:28,280
sense to have as v prime.

252
00:11:28,280 --> 00:11:30,470
So that was the main
point of this exercise.

253
00:11:30,470 --> 00:11:33,110
Hopefully you're starting
to get a flavor for how

254
00:11:33,110 --> 00:11:35,340
these problems actually work.

255
00:11:35,340 --> 00:11:37,331
And I think I will stop there.